Variational Cellular Method Research Articles

Variational cellular model of the energy bands of silicon

The recently proposed variational cellular method is formulated for application to three-dimensional periodic structures with an arbitrary number of atoms per unit cell. The method is used here to determine the electronic structure of silicon. The energy band calculations were carried out with the unit cell partitioned into four space-filling polyhedra. Interstitial cells were included in order to improve on the spherical cellular potentials. The calculations reported show that the method requires a small computational effort to obtain accurate and well-converged results.

The variational cellular method for quantum mechanical applications: Calculations of the ground and excited states of F2 and Ne2 molecules

The recently developed variational cellular method (VCM) is used to investigate the ground and excited electronic structures of F2 and Ne2 molecules. This work reports our first attempt to apply the VCM to the study of molecular excited states. It is concluded that VCM yields potential energy curves for these molecules with a good degree of physical realism.

Variational cellular method in molecular and crystal electronic structure calculations

The variational cellular method (VCM) is applied to calculate the electronic structure of ground-state CO and Na metallic crystal. The results are in fairly good agreement with other calculations and experiment. The free-electron energy eigenvalues in a body-centered-cubic lattice were also obtained. It is found that VCM yields the correct solution to the “empty-lattice test.”

BEYOND DENSITY FUNCTIONAL THEORY: THE DOMESTICATION OF NONLOCAL POTENTIALS

Due to efficient scaling with electron number N, density functional theory (DFT) is widely used for studies of large molecules and solids. Restriction of an exact mean-field theory to local potential functions has recently been questioned. This review summarizes motivation for extending current DFT to include nonlocal one-electron potentials, and proposes methodology for implementation of the theory. The theoretical model, orbital functional theory (OFT), is shown to be exact in principle for the general N-electron problem. In practice it must depend on a parametrized correlation energy functional. Functionals are proposed suitable for short-range Coulomb-cusp correlation and for long-range polarization response correlation. A linearized variational cellular method (LVCM) is proposed as a common formalism for molecules and solids. Implementation of nonlocal potentials is reduced to independent calculations for each inequivalent atomic cell.

Energy‐linearized variational cellular method for large molecules and solids

AbstractMultiple scattering theory (MST) is the mainstay of Kohn–Sham calculations of the electronic structure of solids and alloys. MST formalism solves one‐electron equations within each atomic cell, using a Green function to propagate solutions across cell boundaries, while standard methodology for molecules expands wave functions in a Gaussian orbital basis. An energy‐linearized version of MST (LMTO) is efficient but restricted to an atomic‐sphere model. Full‐potential MST extends the formalism to space‐filling Wigner–Seitz polyhedra. The variational cellular method (VCM) solves full‐potential equations, replacing Green‐function propagation (structure constants) by variational matching at the interfaces of adjacent atomic cells. VCM provides a common formalism for molecules and solids but cannot easily be converted to an energy‐linearized method. A new variational principle is derived here that extends the VCM to a straightforward procedure for energy linearization. This formalism eliminates false solutions from the VCM. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Cellular solutions for the Poisson equation in extended systems.

The Poisson equation for the electrostatic potential in a solid is solved using three different cellular techniques. The relative merits of these different approaches are discussed for two test charge densities for which an analytic solution to the Poisson equation is known. The first approach uses full-cell multiple-scattering theory and results in the famililar structure constant and multipole moment expansion. This solution is shown to be valid everywhere inside the cell, although for points outside the muffin-tin sphere but inside the cell the sums must be performed in the correct order to yield meaningful results. A modification of the multiple-scattering-theory approach yields a second method, a Green-function cellular method, which only requires the solution of a nearest-neighbor linear system of equations. A third approach, a related variational cellular method, is also derived. The variational cellular approach is shown to be the most accurate and reliable, and to have the best convergence in angular momentum of the three methods. Coulomb energies accurate to within ${10}^{\mathrm{\ensuremath{-}}6}$ hartree are easily achieved with the variational cellular approach, demonstrating the practicality of the approach in electronic structure calculations.

Full-potential multiple-scattering theory without structure constants.

A form of full-potential multiple-scattering theory for electrons in solids or molecules has recently been proposed in which the structure constants characteristic of standard theory (Korringa-Kohn-Rostoker) do not occur. This formalism was derived from the Lippmann-Schwinger integral equation and has been called the Green-function cellular method. It is shown here that this formalism is a restatement of the tail-cancellation condition of Andersen, applied originally in the context of his muffin-tin-orbital construction, using a local spherical approximation to the potential function in the Schr\"odinger equation. This was generalized to the atomic-cell orbital (ACO) construction for the full-potential problem by the present author. The equations of this method are derived here directly from the ACO tail-cancellation condition for boundary matching on the surface of each cell in a set of space-filling atomic cells, making no use of the free-particle or Helmholtz Green function. It is also shown here that these equations correspond to a restricted variation of trial functions on the surfaces of atomic cells in the context of the variational cellular method of Leite and collaborators, derived from the variational principle of Schlosser and Marcus.

Internal sums in full-potential multiple-scattering theory.

Full-potential multiple-scattering theory for electrons can be derived from the variational principle of Kohn and Rostoker. The resulting equations differ from the standard Korringa-Kohn-Rostoker (KKR) formalism by a matrix multiplication, which introduces an intermediate sum that should in principle be carried out to completion. This implies that the usual site-diagonal square matrices S and C of KKR theory must be represented as rectangular matrices. In the limit of completeness of this internal sum, the resulting product of matrices can be represented in closed form as a Wronskian integral extended over the common interfaces of adjacent space-filling cells, as used in the variational cellular method (VCM), based on the variational principle of Schlosser and Marcus. The standard KKR method can be extended in two ways, either by direct use of rectangular matrices, or by using the VCM surface integral as a closure correction in the energy-linearized atomic-cell orbital method. Test calculations on an empty-lattice model (fcc three-dimensional space lattice) demonstrate the viability of these extensions.

Cluster calculations of ZnO with Cu and Ni impurities.

The variational cellular method, extended to study crystalline structures, has been applied to 17-atom clusters representing a ZnO crystal without and with substitutional impurities. Self-consistent-field electronic-structure calculations were carried out for the substitutional Cu and Ni. The (2+)-charge ion states were studied and the optical transitions discussed in terms of Koopmans's theorem or transition-state calculations, depending on their range of applicability. The analysis of the results is preceded by a discussion on the general questions of to what extent a cluster can be used to calculate crystal energy bands and localized states.

Band-structure calculations of BN by the self-consistent variational cellular method.

The electronic band-structure calculations of zinc-blende--structure BN is carried out with use of a self-consistent version of the local-density variational cellular method. The crystal potential and charge density are approximated within atomic and interstitial space-filling cells by their spherical average. We find that the values obtained for the valence-band widths and minimum band gaps of BN are in good agreement with the results of recent ab initio calculations.

Linearised variational cellular method

Reviews the variational cellular method for polyatomic systems, and presents its linearised version. This is a very attractive version because of its computational speed. After describing the detailed mathematical foundation of the method, the authors present its application to CO2 as an example.

Ionisation potentials of benzene using the cellular method

The authors use the variational cellular method (VCM) for the calculation of the ionisation potentials (IP) of benzene. The expected precision of the VCM is the same as that of an atomic calculation where charge and potential are spherically averaged. The calculated IP of benzene, if shifted by a constant, are in better agreement with the experiment than those of a Hartree-Fock or a multiple-scattering calculation. The shifted IP have a precision already comparable with those of a configuration interaction calculation. The constant shift of all IP is much too large, but can be understood as resulting from the assumed full symmetry of the molecular ion.

Band structure of silicon by the self-consistent variational cellular method

The self-consistent formulation of the variational cellular method has been developed in order to calculate the electronic structure of crystals with an arbitrary number of atoms per unit cell. Applications for silicon have been carried out. Silicon, chosen here as a test case, is treated as a face-centered cubic lattice with four “atoms” per unit cell by adding empty cells. The electronic charge density was taken muffin-tin, assuming a constant value in the interstitial region between the inscribed sphere and the Wigner–Seitz polyhedrum. The spherical symmetric electronic charge density in the inscribed sphere was obtained by adding a limited number of contributions of Brillouin zone states using the “mean value point theory” developed by Baldereschi and Chadi-Cohen. Our results are in good agreement with those obtained by other methods.

Self-consistent formulation of the variational cellular method applied to periodic structures—results for sodium and silicon

The theoretical formulation of the self-consistent variational cellular method (SCVCM) has been developed in order to calculate the electronic structure of crystals with an arbitrary number of atoms per unit cell. Calculations for metallic sodium and silicon have been carried out. The electronic charge density within the atomic polyhedron was calculated assuming two regions, one corresponding to the inscribed sphere and the other corresponding to the interstitial region. The radial electronic charge density in the inscribed sphere was obtained by adding a limited number of contributions of Brillouin zone states using the “mean value point theory”, developed by Baldereschi and Chadi-Cohen. In the interstitial region, the electronic density was taken as of a constant value. The model can be extended to open structures by adding empty cells. Our results for sodium and silicon are in good agreement with those obtained by other methods. The self-consistent scheme proposed is accurate and fast enough to be applied to more complex structures.

Application of the variational cellular method to semiconductors: The ZnS case

We have developed a new version of the variational cellular method for the calculation of the electronic structure of polyatomic systems. It is a very attractive method for large systems, such as solids and large molecules, and capable of giving calculated results with a degree of precision at least as good as those obtained by Hartree–Fock calculations. We are now improving the method by the introduction of crystalline boundary conditions with the intention to apply it to the study of impurities in semiconductors. The results for the ZnS clusters simulating the pure crystal show that the method leads to a fairly consistent description of the bulk electronic structure.

Variational cellular method for polyatomic molecules: SiF4and SiCl4

Presents the new developments in the theory and applications of the variational cellular method. The most important advance in the theory was the introduction of a continuity condition for the wavefunctions and potential at the cell boundaries. The continuity condition gave a very remarkable stability to the results for changes in the size of the angular momentum expansions. The calculated molecules were SiF4 and SiCl4, for which the ionisation potentials are better than those of the multiple-scattering or existing ab initio methods.

Variational cellular method for polyatomic molecules: SF6

AbstractThis is the third paper on the cellular method for polyatomic systems. We show how to deal with nonspherical Coulomb potentials. We also show how to modify the variational expression for the energy eigenvalues so as to obtain a faster convergence in the angular momentum series for the wavefunctions. We apply both techniques to the self‐consistent calculation of SF6. Contrary to what we obtained in CH4 and SiH4, the cellular method cannot yield the correct equilibrium interatomic distance in the present case. The calculated ionization potentials are in the correct order but are all shifted by 2–3 eV. This shift is attributed to the wrong expression for exchange correlation.

Study of some electronic properties of the BH −4 ion through the variational cellular method

We study some properties of the BH − 4 ion through the recently developed variational cellular method for polyatomic systems. The total energy, the B-H equilibrium distance, the breathing mode frequency and SCF vertical ionization potentials (VIP's) are calculated. Some approximated calculation for the VIP's are also made, being suggested a prescription for rapid and precise estimations. We also perform calculations including a Watson sphere, being shown that its effects can be removed from the total energy and the VIP's calculations. A charge analysis indicates a charge decreasing near the hydrogen sites, relative to the atomic charge density.

Variational cellular model of the energy bands of diamond and silicon

We extend the previously proposed variational cellular method to three-dimensional periodic structures with an arbitrary number of atoms per unit cell. The method is used to determine the energy bands of diamond and silicon. The calculations were carried out within the framework of the spherical-cellular-potential approximation with the unit cell partitioned into four space-filling polyhedra. The results obtained are in fairly good agreement with orthogonalized-plane-wave calculations and with observed photoemission and optical data.

Variational cellular method for polyatomic molecules: SiH4

We review the variational cellular method for polyatomic systems and discuss some of its most interesting features. This method may be as fast as the multiple scattering method but is free from the defects of the muffin-tin potential and charge density. Thus it is a method very attractive for large systems such as solids and large molecules, and capable of giving calculated results with a degree of precision at least as good as those of Hartree–Fock calculations. The method is applied to SiH4, where we explore its capabilities with excellent results.